Theorem iunin2 4031

Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4020 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
iunin2	|- U_x e. A (B i^i C) = (B i^i U_x e. A C)

Distinct variable group:   x,B

Allowed substitution hints:   A(x)   C(x)

Proof of Theorem iunin2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.42v 2766	. . . 4 |- (E.x e. A (y e. B /\ y e. C) <-> (y e. B /\ E.x e. A y e. C))
2		elin 3220	. . . . 5 |- (y e. (B i^i C) <-> (y e. B /\ y e. C))
3	2	rexbii 2640	. . . 4 |- (E.x e. A y e. (B i^i C) <-> E.x e. A (y e. B /\ y e. C))
4		eliun 3974	. . . . 5 |- (y e. U_x e. A C <-> E.x e. A y e. C)
5	4	anbi2i 675	. . . 4 |- ((y e. B /\ y e. U_x e. A C) <-> (y e. B /\ E.x e. A y e. C))
6	1, 3, 5	3bitr4i 268	. . 3 |- (E.x e. A y e. (B i^i C) <-> (y e. B /\ y e. U_x e. A C))
7		eliun 3974	. . 3 |- (y e. U_x e. A (B i^i C) <-> E.x e. A y e. (B i^i C))
8		elin 3220	. . 3 |- (y e. (B i^i U_x e. A C) <-> (y e. B /\ y e. U_x e. A C))
9	6, 7, 8	3bitr4i 268	. 2 |- (y e. U_x e. A (B i^i C) <-> y e. (B i^i U_x e. A C))
10	9	eqriv 2350	1 |- U_x e. A (B i^i C) = (B i^i U_x e. A C)

Syntax hints:   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616   i^i cin 3209  U_ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-iun 3972

This theorem is referenced by:  iunin1  4032  2iunin  4035